1033: Canonical 'Vectors Spaces - Linear Morphisms' Isomorphism Between Finite-Dimensional Vectors Space and Its Double Dual Space

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definition of canonical 'vectors spaces - linear morphisms' isomorphism between finite-dimensional vectors space and its double dual space

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of dual basis for covectors (dual) space of basis for finite-dimensional vectors space.
• The reader knows a definition of %category name% isomorphism.

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Target Context

• The reader will have a definition of canonical 'vectors spaces - linear morphisms' isomorphism between finite-dimensional vectors space and its double dual space.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$F$: $\in \{\text{ the fields }\}$
$V$: $\in \{\text{ the finite-dimensional }F\text{ vectors spaces }\}$
$V^{\ast }$: $=L(V:F)$
${V^{\ast }}^{\ast }$: $=L(L(V:F):F)$
$J$: $\in \{\text{ the finite index sets }\}$
$B$: $\in \{\text{ the bases for }V\}$, $=\{b_j|j\in J\}$
$B^{\ast }$: $=\text{ the dual basis of }B$, $=\{b^j|j\in J\}$
${B^{\ast }}^{\ast }$: $=\text{ the dual basis of }{B}^{\ast }$, $=\{{\tilde{b}}_j|j\in J\}$
$\ast f$: $:V\to {V^{\ast }}^{\ast },v^j{b}_j\mapsto \sum _{j\in J}{v}^j{\tilde{b}}_j$, $\in \{\text{ the 'vectors spaces - linear morphisms' isomorphisms }\}$
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Conditions:
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$f$ does not depend on the choice of $B$: compare to the definition of canonical 'vectors spaces - linear morphisms' isomorphism between finite-dimensional vectors space and its covectors space with respect to original space basis with "with respect to original space basis".

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2: Note

$f$ is indeed a 'vectors spaces - linear morphisms' isomorphism, does not depend on the choice of $B$, and has the property that for each $v\in V$ and each $w\in V^{\ast }$, $w(v)=f(v)(w)$, by the proposition that between any finite-dimensional vectors space and its double dual, there is the canonical 'vectors spaces - linear morphisms' isomorphism.

Although it is sometimes sloppily expressed like "The double dual of a finite-dimensional vectors space is the original vectors space.", the double dual is not the same entity with the original vectors space, as the 2 have different meanings. They are just 'vectors spaces - linear morphisms' isomorphic, and having such a relation does not make 2 entities the same.

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References

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